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Properties

Label 2-464-116.115-c2-0-21
Degree $2$
Conductor $464$
Sign $0.510 + 0.859i$
Analytic cond. $12.6430$
Root an. cond. $3.55571$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.231·3-s + 4.30·5-s − 6.80i·7-s − 8.94·9-s + 19.6·11-s − 7.70·13-s + 0.994·15-s − 21.6i·17-s + 7.18·19-s − 1.57i·21-s − 9.72i·23-s − 6.47·25-s − 4.14·27-s + (14.1 − 25.2i)29-s − 3.38·31-s + ⋯
L(s)  = 1  + 0.0770·3-s + 0.860·5-s − 0.971i·7-s − 0.994·9-s + 1.78·11-s − 0.592·13-s + 0.0663·15-s − 1.27i·17-s + 0.378·19-s − 0.0748i·21-s − 0.422i·23-s − 0.259·25-s − 0.153·27-s + (0.489 − 0.872i)29-s − 0.109·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 + 0.859i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $0.510 + 0.859i$
Analytic conductor: \(12.6430\)
Root analytic conductor: \(3.55571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :1),\ 0.510 + 0.859i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.967035249\)
\(L(\frac12)\) \(\approx\) \(1.967035249\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (-14.1 + 25.2i)T \)
good3 \( 1 - 0.231T + 9T^{2} \)
5 \( 1 - 4.30T + 25T^{2} \)
7 \( 1 + 6.80iT - 49T^{2} \)
11 \( 1 - 19.6T + 121T^{2} \)
13 \( 1 + 7.70T + 169T^{2} \)
17 \( 1 + 21.6iT - 289T^{2} \)
19 \( 1 - 7.18T + 361T^{2} \)
23 \( 1 + 9.72iT - 529T^{2} \)
31 \( 1 + 3.38T + 961T^{2} \)
37 \( 1 - 41.4iT - 1.36e3T^{2} \)
41 \( 1 + 24.1iT - 1.68e3T^{2} \)
43 \( 1 - 59.7T + 1.84e3T^{2} \)
47 \( 1 - 51.6T + 2.20e3T^{2} \)
53 \( 1 + 29.5T + 2.80e3T^{2} \)
59 \( 1 + 36.7iT - 3.48e3T^{2} \)
61 \( 1 + 91.5iT - 3.72e3T^{2} \)
67 \( 1 + 80.0iT - 4.48e3T^{2} \)
71 \( 1 + 112. iT - 5.04e3T^{2} \)
73 \( 1 - 88.8iT - 5.32e3T^{2} \)
79 \( 1 + 57.8T + 6.24e3T^{2} \)
83 \( 1 - 132. iT - 6.88e3T^{2} \)
89 \( 1 - 24.3iT - 7.92e3T^{2} \)
97 \( 1 - 116. iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.66550026398978371375337655132, −9.544175779582783402876281359603, −9.239848125160519049022418959915, −7.918682250369661414762050827660, −6.86517192889740886441910149828, −6.11143936453907664638167320552, −4.92541189459274842517940669832, −3.75937874222748942795524830437, −2.42481917295085004773396555375, −0.854301751948601171705431919378, 1.58596764754452707958041652444, 2.76240517924315300284888781891, 4.11231188093239490537970306310, 5.71784335302074235538085233822, 5.95333682413006542695431200651, 7.21963158620123840895862024005, 8.665701317542073098173588302599, 9.048621906908571159287195266603, 9.910628788851322023256202499473, 11.06148947514121529034351366623

Graph of the $Z$-function along the critical line